The ratio of the rise and the run, called the difference quotient, that equals the value of the tangent of the angle between the direction of the secant line and x-axis, becomes the slope (gradient) of the tangent line as the difference Dx tends to zero, and is called the instantaneous rate of change or the derivative at the point of the function.
For a given function ƒ and point (x1, ƒ (x1)), the derivative of ƒ at x = x1 is the slope of the tangent line through the point (x1, ƒ (x1)), i.e., f '(x1) = tan at
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point